Geometry and Mathematical Physics Seminar 2007/2008

Given a vector bundle E over a manifold M and a symmetric bilinear form <,> on E, we construct a graded commutative algebra whose element in degree n are n-ary operations on sections of E with values in functions on M. These operations are neither multilinear over functions on M nor skew-symmetric; rather, their behavior under permutations of the arguments and multiplication by functions is controlled by <,>. If <,> is non-degenerate, there is also a graded non-degenerate Poisson bracket of degree -2 on the algebra, such that solutions of the Maurer-Cartan equation correspond precisely to Courant algebroid structures on (E,<,>). This algebra is isomorphic to the algebra of functions on a graded symplectic supermanifold for which there is a simple geometric construction. The construction of the algebra can also be done in the setting of modules over a commutative ring, yielding a deformation complex for Courant-Dorfman algebras.